39:
1437:
1153:
530:
1167:
877:
862:
628:
Since three or fewer points are always coplanar, the problem of determining when a set of points are coplanar is generally of interest only when there are at least four points involved. In the case that there are exactly four points, several
428:
1432:{\displaystyle {\begin{bmatrix}x_{1}-w_{1}&x_{2}-w_{2}&\dots &x_{n}-w_{n}\\y_{1}-w_{1}&y_{2}-w_{2}&\dots &y_{n}-w_{n}\\z_{1}-w_{1}&z_{2}-w_{2}&\dots &z_{n}-w_{n}\\\end{bmatrix}}}
733:
882:
1148:{\displaystyle {\begin{aligned}X&=(x_{1},x_{2},\dots ,x_{n}),\\Y&=(y_{1},y_{2},\dots ,y_{n}),\\Z&=(z_{1},z_{2},\dots ,z_{n}),\\W&=(w_{1},w_{2},\dots ,w_{n}),\end{aligned}}}
380:
252:
564:
738:
633:
methods can be employed, but a general method that works for any number of points uses vector methods and the property that a plane is determined by two
525:{\displaystyle (\mathbf {c} \cdot \mathbf {\hat {a}} )\mathbf {\hat {a}} +(\mathbf {c} \cdot \mathbf {\hat {b}} )\mathbf {\hat {b}} =\mathbf {c} ,}
68:
provides a solution technique for the problem of determining whether a set of points is coplanar, knowing only the distances between them.
735:
are coplanar if and only if the matrix of their relative differences, that is, the matrix whose columns (or rows) are the vectors
35:, the plane they determine is unique. However, a set of four or more distinct points will, in general, not lie in a single plane.
92:
to this cross product through the initial point will lie in the plane. This leads to the following coplanarity test using a
658:
1445:
In the special case of a plane that contains the origin, the property can be simplified in the following way: A set of
1535:
50:
in three-dimensional space are coplanar if there is a plane that includes them both. This occurs if the lines are
263:
135:
857:{\overrightarrow {p_{0}p_{1}}},\ {\overrightarrow {p_{0}p_{2}}},\ \dots ,\ {\overrightarrow {p_{0}p_{k-1}}}
634:
31:
that contains them all. For example, three points are always coplanar, and if the points are distinct and
540:
1582:
1473:
are not coplanar. Such a polygon must have at least four vertices; there are no skew triangles.
55:
1527:
1520:
865:
93:
1159:
573:
77:
32:
8:
51:
28:
1498:
1449:
points and the origin are coplanar if and only if the matrix of the coordinates of the
1555:
1531:
1470:
583:
85:
65:
619:
47:
80:
vectors with the same initial point determine a plane through that point. Their
1558:
1576:
81:
1493:
1462:
569:
1477:
89:
59:
1563:
20:
38:
1466:
1481:
1526:(Alternate ed.), Prindle, Weber & Schmidt, p.
58:
each other. Two lines that are not coplanar are called
1176:
1170:
880:
661:
543:
431:
266:
138:
16:
Geometric property of objects being in the same plane
1442:
is of rank 2 or less, the four points are coplanar.
728:{\displaystyle \{p_{0},\ p_{1},\ \dots ,\ p_{k-1}\}}
1519:
1431:
1147:
856:
727:
558:
524:
374:
246:
1553:
550:
505:
490:
464:
449:
71:
1574:
722:
662:
1517:
1484:has vertices that are not all coplanar.
37:
1575:
624:dimensions whose coordinates are given
375:{\displaystyle (x_{2}-x_{1})\cdot =0.}
247:{\displaystyle \cdot (x_{3}-x_{1})=0.}
1554:
88:vector to that plane, and any vector
1456:
559:{\displaystyle \mathbf {\hat {a}} }
13:
14:
1594:
1547:
547:
515:
502:
487:
477:
461:
446:
436:
76:In three-dimensional space, two
1522:Calculus with Analytic Geometry
871:For example, given four points
129:, are coplanar if and only if,
23:, a set of points in space are
1511:
1135:
1090:
1070:
1025:
1005:
960:
940:
895:
496:
473:
455:
432:
363:
360:
334:
328:
302:
299:
293:
267:
235:
209:
203:
200:
174:
168:
142:
139:
72:Properties in three dimensions
1:
1504:
1453:points is of rank 2 or less.
42:An example of coplanar points
635:linearly independent vectors
257:which is also equivalent to
27:if there exists a geometric
7:
1518:Swokowski, Earl W. (1983),
1487:
10:
1599:
644:-dimensional space where
620:Coplanarity of points in
610:add to give the original
1433:
1149:
858:
729:
560:
526:
399:are coplanar, then if
376:
248:
99:Four distinct points,
43:
1434:
1150:
859:
730:
561:
527:
422:are orthogonal) then
377:
249:
94:scalar triple product
41:
1168:
878:
739:
659:
541:
429:
264:
136:
78:linearly independent
1556:Weisstein, Eric W.
1499:Plane of incidence
1480:that has positive
1429:
1423:
1145:
1143:
725:
584:vector projections
556:
522:
372:
244:
44:
1583:Planes (geometry)
853:
819:
810:
803:
775:
768:
705:
696:
680:
553:
508:
493:
467:
452:
385:If three vectors
66:Distance geometry
1590:
1569:
1568:
1541:
1540:
1525:
1515:
1457:Geometric shapes
1452:
1448:
1438:
1436:
1435:
1430:
1428:
1427:
1420:
1419:
1407:
1406:
1390:
1389:
1377:
1376:
1365:
1364:
1352:
1351:
1338:
1337:
1325:
1324:
1308:
1307:
1295:
1294:
1283:
1282:
1270:
1269:
1256:
1255:
1243:
1242:
1226:
1225:
1213:
1212:
1201:
1200:
1188:
1187:
1154:
1152:
1151:
1146:
1144:
1134:
1133:
1115:
1114:
1102:
1101:
1069:
1068:
1050:
1049:
1037:
1036:
1004:
1003:
985:
984:
972:
971:
939:
938:
920:
919:
907:
906:
863:
861:
860:
855:
854:
849:
848:
847:
832:
831:
821:
817:
808:
804:
799:
798:
797:
788:
787:
777:
773:
769:
764:
763:
762:
753:
752:
742:
734:
732:
731:
726:
721:
720:
703:
694:
690:
689:
678:
674:
673:
654:
650:
643:
615:
609:
603:
597:
591:
581:
567:
565:
563:
562:
557:
555:
554:
546:
531:
529:
528:
523:
518:
510:
509:
501:
495:
494:
486:
480:
469:
468:
460:
454:
453:
445:
439:
421:
415:
409:
398:
381:
379:
378:
373:
359:
358:
346:
345:
327:
326:
314:
313:
292:
291:
279:
278:
253:
251:
250:
245:
234:
233:
221:
220:
199:
198:
186:
185:
167:
166:
154:
153:
128:
1598:
1597:
1593:
1592:
1591:
1589:
1588:
1587:
1573:
1572:
1550:
1545:
1544:
1538:
1516:
1512:
1507:
1490:
1459:
1450:
1446:
1422:
1421:
1415:
1411:
1402:
1398:
1396:
1391:
1385:
1381:
1372:
1368:
1366:
1360:
1356:
1347:
1343:
1340:
1339:
1333:
1329:
1320:
1316:
1314:
1309:
1303:
1299:
1290:
1286:
1284:
1278:
1274:
1265:
1261:
1258:
1257:
1251:
1247:
1238:
1234:
1232:
1227:
1221:
1217:
1208:
1204:
1202:
1196:
1192:
1183:
1179:
1172:
1171:
1169:
1166:
1165:
1142:
1141:
1129:
1125:
1110:
1106:
1097:
1093:
1083:
1077:
1076:
1064:
1060:
1045:
1041:
1032:
1028:
1018:
1012:
1011:
999:
995:
980:
976:
967:
963:
953:
947:
946:
934:
930:
915:
911:
902:
898:
888:
881:
879:
876:
875:
837:
833:
827:
823:
822:
820:
793:
789:
783:
779:
778:
776:
758:
754:
748:
744:
743:
741:
740:
737:
736:
710:
706:
685:
681:
669:
665:
660:
657:
656:
652:
645:
641:
626:
611:
605:
599:
593:
587:
582:. That is, the
577:
545:
544:
542:
539:
538:
536:
514:
500:
499:
485:
484:
476:
459:
458:
444:
443:
435:
430:
427:
426:
417:
411:
400:
386:
354:
350:
341:
337:
322:
318:
309:
305:
287:
283:
274:
270:
265:
262:
261:
229:
225:
216:
212:
194:
190:
181:
177:
162:
158:
149:
145:
137:
134:
133:
127:
120:
113:
106:
100:
74:
17:
12:
11:
5:
1596:
1586:
1585:
1571:
1570:
1549:
1548:External links
1546:
1543:
1542:
1536:
1509:
1508:
1506:
1503:
1502:
1501:
1496:
1489:
1486:
1458:
1455:
1440:
1439:
1426:
1418:
1414:
1410:
1405:
1401:
1397:
1395:
1392:
1388:
1384:
1380:
1375:
1371:
1367:
1363:
1359:
1355:
1350:
1346:
1342:
1341:
1336:
1332:
1328:
1323:
1319:
1315:
1313:
1310:
1306:
1302:
1298:
1293:
1289:
1285:
1281:
1277:
1273:
1268:
1264:
1260:
1259:
1254:
1250:
1246:
1241:
1237:
1233:
1231:
1228:
1224:
1220:
1216:
1211:
1207:
1203:
1199:
1195:
1191:
1186:
1182:
1178:
1177:
1175:
1156:
1155:
1140:
1137:
1132:
1128:
1124:
1121:
1118:
1113:
1109:
1105:
1100:
1096:
1092:
1089:
1086:
1084:
1082:
1079:
1078:
1075:
1072:
1067:
1063:
1059:
1056:
1053:
1048:
1044:
1040:
1035:
1031:
1027:
1024:
1021:
1019:
1017:
1014:
1013:
1010:
1007:
1002:
998:
994:
991:
988:
983:
979:
975:
970:
966:
962:
959:
956:
954:
952:
949:
948:
945:
942:
937:
933:
929:
926:
923:
918:
914:
910:
905:
901:
897:
894:
891:
889:
887:
884:
883:
852:
846:
843:
840:
836:
830:
826:
816:
813:
807:
802:
796:
792:
786:
782:
772:
767:
761:
757:
751:
747:
724:
719:
716:
713:
709:
702:
699:
693:
688:
684:
677:
672:
668:
664:
625:
618:
552:
549:
533:
532:
521:
517:
513:
507:
504:
498:
492:
489:
483:
479:
475:
472:
466:
463:
457:
451:
448:
442:
438:
434:
383:
382:
371:
368:
365:
362:
357:
353:
349:
344:
340:
336:
333:
330:
325:
321:
317:
312:
308:
304:
301:
298:
295:
290:
286:
282:
277:
273:
269:
255:
254:
243:
240:
237:
232:
228:
224:
219:
215:
211:
208:
205:
202:
197:
193:
189:
184:
180:
176:
173:
170:
165:
161:
157:
152:
148:
144:
141:
125:
118:
111:
104:
73:
70:
15:
9:
6:
4:
3:
2:
1595:
1584:
1581:
1580:
1578:
1566:
1565:
1560:
1557:
1552:
1551:
1539:
1537:0-87150-341-7
1533:
1529:
1524:
1523:
1514:
1510:
1500:
1497:
1495:
1492:
1491:
1485:
1483:
1479:
1474:
1472:
1468:
1464:
1454:
1443:
1424:
1416:
1412:
1408:
1403:
1399:
1393:
1386:
1382:
1378:
1373:
1369:
1361:
1357:
1353:
1348:
1344:
1334:
1330:
1326:
1321:
1317:
1311:
1304:
1300:
1296:
1291:
1287:
1279:
1275:
1271:
1266:
1262:
1252:
1248:
1244:
1239:
1235:
1229:
1222:
1218:
1214:
1209:
1205:
1197:
1193:
1189:
1184:
1180:
1173:
1164:
1163:
1162:
1161:
1138:
1130:
1126:
1122:
1119:
1116:
1111:
1107:
1103:
1098:
1094:
1087:
1085:
1080:
1073:
1065:
1061:
1057:
1054:
1051:
1046:
1042:
1038:
1033:
1029:
1022:
1020:
1015:
1008:
1000:
996:
992:
989:
986:
981:
977:
973:
968:
964:
957:
955:
950:
943:
935:
931:
927:
924:
921:
916:
912:
908:
903:
899:
892:
890:
885:
874:
873:
872:
869:
867:
850:
844:
841:
838:
834:
828:
824:
814:
811:
805:
800:
794:
790:
784:
780:
770:
765:
759:
755:
749:
745:
717:
714:
711:
707:
700:
697:
691:
686:
682:
675:
670:
666:
648:
638:
636:
632:
623:
617:
614:
608:
602:
596:
590:
585:
580:
575:
571:
519:
511:
481:
470:
440:
425:
424:
423:
420:
414:
407:
403:
397:
393:
389:
369:
366:
355:
351:
347:
342:
338:
331:
323:
319:
315:
310:
306:
296:
288:
284:
280:
275:
271:
260:
259:
258:
241:
238:
230:
226:
222:
217:
213:
206:
195:
191:
187:
182:
178:
171:
163:
159:
155:
150:
146:
132:
131:
130:
124:
117:
110:
103:
97:
95:
91:
87:
83:
82:cross product
79:
69:
67:
63:
61:
57:
54:, or if they
53:
49:
40:
36:
34:
33:non-collinear
30:
26:
22:
1562:
1521:
1513:
1494:Collinearity
1475:
1463:skew polygon
1460:
1444:
1441:
1157:
870:
868:2 or less.
646:
639:
630:
627:
621:
612:
606:
600:
594:
588:
578:
568:denotes the
534:
418:
412:
405:
401:
395:
391:
387:
384:
256:
122:
115:
108:
101:
98:
75:
64:
45:
24:
18:
651:, a set of
570:unit vector
1559:"Coplanar"
1505:References
1478:polyhedron
90:orthogonal
60:skew lines
1564:MathWorld
1409:−
1394:…
1379:−
1354:−
1327:−
1312:…
1297:−
1272:−
1245:−
1230:…
1215:−
1190:−
1120:…
1055:…
990:…
925:…
851:→
842:−
812:…
801:→
766:→
715:−
698:…
574:direction
551:^
506:^
491:^
482:⋅
465:^
450:^
441:⋅
348:−
332:×
316:−
297:⋅
281:−
223:−
207:⋅
188:−
172:×
156:−
56:intersect
1577:Category
1488:See also
1471:vertices
52:parallel
25:coplanar
21:geometry
1467:polygon
1158:if the
655:points
572:in the
566:
537:
410:(i.e.,
1534:
1482:volume
1469:whose
1160:matrix
864:is of
818:
809:
774:
704:
695:
679:
640:In an
631:ad hoc
535:where
86:normal
1465:is a
84:is a
48:lines
29:plane
1532:ISBN
866:rank
598:and
416:and
46:Two
1528:647
649:≥ 3
604:on
592:on
586:of
576:of
408:= 0
19:In
1579::
1561:.
1530:,
1476:A
1461:A
637:.
616:.
404:⋅
394:,
390:,
370:0.
242:0.
121:,
114:,
107:,
96::
62:.
1567:.
1451:k
1447:k
1425:]
1417:n
1413:w
1404:n
1400:z
1387:2
1383:w
1374:2
1370:z
1362:1
1358:w
1349:1
1345:z
1335:n
1331:w
1322:n
1318:y
1305:2
1301:w
1292:2
1288:y
1280:1
1276:w
1267:1
1263:y
1253:n
1249:w
1240:n
1236:x
1223:2
1219:w
1210:2
1206:x
1198:1
1194:w
1185:1
1181:x
1174:[
1139:,
1136:)
1131:n
1127:w
1123:,
1117:,
1112:2
1108:w
1104:,
1099:1
1095:w
1091:(
1088:=
1081:W
1074:,
1071:)
1066:n
1062:z
1058:,
1052:,
1047:2
1043:z
1039:,
1034:1
1030:z
1026:(
1023:=
1016:Z
1009:,
1006:)
1001:n
997:y
993:,
987:,
982:2
978:y
974:,
969:1
965:y
961:(
958:=
951:Y
944:,
941:)
936:n
932:x
928:,
922:,
917:2
913:x
909:,
904:1
900:x
896:(
893:=
886:X
845:1
839:k
835:p
829:0
825:p
815:,
806:,
795:2
791:p
785:0
781:p
771:,
760:1
756:p
750:0
746:p
723:}
718:1
712:k
708:p
701:,
692:,
687:1
683:p
676:,
671:0
667:p
663:{
653:k
647:n
642:n
622:n
613:c
607:b
601:c
595:a
589:c
579:a
548:a
520:,
516:c
512:=
503:b
497:)
488:b
478:c
474:(
471:+
462:a
456:)
447:a
437:c
433:(
419:b
413:a
406:b
402:a
396:c
392:b
388:a
367:=
364:]
361:)
356:1
352:x
343:3
339:x
335:(
329:)
324:1
320:x
311:4
307:x
303:(
300:[
294:)
289:1
285:x
276:2
272:x
268:(
239:=
236:)
231:1
227:x
218:3
214:x
210:(
204:]
201:)
196:1
192:x
183:4
179:x
175:(
169:)
164:1
160:x
151:2
147:x
143:(
140:[
126:4
123:x
119:3
116:x
112:2
109:x
105:1
102:x
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
